Preheat oven to 325 degrees, with rack in the lower third of oven. Sift together flour and 3/4 cup sugar 4 times.
Beat egg whites and warm water with a mixer on low speed until foamy, about 3 minutes. Add cream of tartar, vanilla, salt, and coconut extract. Raise speed to medium, and beat until soft peaks form, about 3 minutes. With machine running, add remaining 3/4 cup sugar, 1 tablespoon at a time. Raise speed to high, and beat until peaks are stiff and glossy, but not dry, about 2 minutes.
Working in 3 batches, sift flour mixture over egg-white mixture, gently folding in each addition with a rubber spatula, being careful not to deflate egg whites.
Slowly pour batter into a 10-inch angel food cake pan. Gently run a knife through center of batter to release any air bubbles, and smooth using an offset spatula. Bake until cake is golden brown and springs back when touched, 40 to 45 minutes.
Invert pan onto a wire rack. Let cake cool in pan, about 1 hour. Carefully run a knife around sides of cake to loosen, and turn out onto a large plate or cake stand.
Pour coconut glaze onto cake, and spread evenly using an offset spatula to cover completely. Gently press coconut onto cake. Let stand until set, about 1 hour. Garnish with lollipops.
Top with Spun Sugar instead of coconut if desired.
To make eight 4-inch mini angel food cakes, follow instructions through step 3. Transfer batter to a pastry bag, and pipe each mold (bakerstools.com) three-quarters full. Bake for 25 to 30 minutes. You will need a double batch of glaze to pour over the cakes.